DiscontinuityPlot is used to plot real-valued discontinuous functions of one real variable, such as the tangent. At the discontinuity, the second derivative of the function assumes very high absolute values. This is used to locate the discontinuities and to omit them from the plot. You can use the Threshold-option in order to give the threshold value for the second derivative (if the second derivative exceeds the threshold value at a particular point, the point is omitted from the plot).